Brad, I'm glad to learn the Lasik went well and is behind you. That is good news.
Triumphs were notorious for electrical problems - Meridan Triumphs back in "the day". The modern Hinckley Triumphs do not have a reputation for electrical problems. Todays electrical systems have more relays, safety switches, etc by far than bikes did thirty years ago and they have far fewer problems than bikes did thirty years ago. The old Lucas electrics of the classic Triumph era were notorious for problems - and that extended to British cars: MG's Jags, etc. Having owned both of these brands in the 70's, I can personally vouch for that. Todays Triumphs have no more electrical problems than other brands and less than some.
That said, the modern electrical systems are complicate and time consuming to diagnose when something does go wrong be it a Triumph, a BMW, or a Cadillac. All of the lighting and the ignition switch wiring on Triumph Bonneville family run through the headlight shell. Because of the "lights on" when the key is on mandate, the lights are tied to the ignition switch. All of this complicates tracing a wiring problem. I have 30 K miles on my Bonnie and Mae Lyne has 21K on hers. Both bikes have been parked outside, ridden in driving rain multiple times and washed frequently. Neither bike has had an electrical problem except for me pinching a wire in mine causing it to ground. The only thing I have done electrically is replace headlight bulbs twice in each bike.
Known electrical problems (and very infrequent) for the Bonnie family are the kickstand and neutral switches. the pick up sensor coil, and moisture getting in the connectors. I had moisture get in a connection in a rain storm one time causing the bike to start missing. As I slowed down and pulled over, the engine heat dried it out and it ran fine. Dielectric grease has negated a repeat performance.
I regret that you got a bike with an electrical problem. It is frustrating and a big inconvenience when a brand new bike or automobile has a problem. The upside is that it will be fixed under warranty and when it is fixed properly, you wlil have no more problems of that nature.
Please keep us posted on what they do.